TITLE: Definition of forgetful functor
QUESTION [9 upvotes]: Is there an actual definition of "forgetful functor?" Wolfram MathWorld defines it in terms of functors from algebraic categories to the category of sets, but then says, "Other forgetful functors..." which seems an odd way to "define" something. In any event, it seems to exclude, say, the functor which takes topological groups to topological spaces (or groups.)
Wikipedia contains no definition, just lots of examples.
Is this one of those cases where we don't need a strict definition, we just recognize it when we see it?

REPLY [7 votes]: I think that calling a functor forgetful is making a choice of a particular perspective on the source category, rather than declaring that the functor has some property. 
When you choose to think of a functor $F : C \to D$ as forgetful, you're choosing to think of the objects in $C$ as objects in $D$ with extra structure in some vague sense. After all, from the perspective of abstract category theory $C$ is some opaque collection of objects and morphisms, which have no internal structure except what internal structure you choose to provide them, e.g. by choosing suitable forgetful functors.
(Incidentally, this is also an answer to a different question that one might want to ask about category theory, namely "what's the point of objects if all of the structure of a category is encoded in its morphisms?" One answer is that how you choose to describe the objects of a category can be thought of as declaring an intent to study a particular collection of forgetful functors out of your category. For example, choosing to describe the objects as sets with extra structure corresponds to declaring an intent to study a particular forgetful functor to $\text{Set}$.)
"Faithful functor" is not a terrible approximation but let me point out one example where it doesn't apply: many people would be happy to say that there is a forgetful functor from schemes to topological spaces, but this functor fails to be faithful. From my perspective, though, all that people mean when they say this is that they're declaring an intent to study schemes as topological spaces with extra structure (namely the structure sheaf).